 Hi, well, I'm Professor Nesheba, and I'm here to tell you a little bit about the question Why is that integral equal to the work in thermodynamics? So to get at this, I'm just going to break out the integrand or the part that's inside the integral which says that minus the external pressure times the change in volume is equal to a little bit of a little bit of work and and for a context let's just imagine you have a beaker and There's a there's a maybe a chemical reaction that's going on inside a beaker that's producing bubbles and you can just imagine that these bubbles are coming out and if the you know, they could you can imagine that they are just sort of forming a layer of what that gas was slowly moving up and pushing away the pushing away the atmosphere and So let's talk a little bit about How much volume that is? In here that that we just pushed up. Well, I could say that that volume is Is equal to a small distance. I'll call it DZ that we that imaginary disc got pushed up and There's certain area that our beaker covers and of course then that means that the volume that were rather the change in volume must be equal to the area times that that distance that's for area of this little wafer and So that's the change in volume So that's what I've written right here that the change in volume is equal to a times times DZ Now what about the pressure? This is the pressure of the of the of the surrounding pushing down on that that this gas has to push away and Well pressure is just a force per unit area that same area So I can write this same expression the little bit of work that's being done is equal to the external pressure force divided by area Times the little change in volume area times DZ and obviously the areas are going to cancel out And so we have that little bit of work is equal to the negative of the force times distances So we have sort of like the conventional physics expression that work is a force through a distance and to get to the you know the work that would have been done over a you know a longer period and taking into account the possibility that maybe that external pressure could change as As these bubbles are creeping up then what we would do is we would say that's equal to the integral of the external pressure over Increases in volume with that minus sign in front. So that's why the integral of Px-ternal dB is is called the work in thermodynamics